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As this is a list-making question I am converting it to Community wiki consistent with our policy on reference requests and the like. I'll be opening a topic on meta concerning this whole class of questions shortly.
–
dmckee♦Nov 29 '11 at 16:09

2 Answers
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Here is one I just made up, but it has a nice flavor--- suppose you have a 2-d bullet going very fast through a 2-d gas. The gas molecules reflects specularly off the bullet, making glancing collisions. What shape of bullet of a fixed area has the least drag?

This problem gives

$$\int {1\over 1+y'^2} + \lambda y dx $$

And the equation for y' you get is

$$ y' = \lambda x (1 - 2 y'^2 - y'^4) $$

or

$$ y = \int \lambda x ( 1 - 2 y^2 - y^4) $$

Which you can solve in a series by plugging in $y={\lambda x^2\over 2}$ and iterating a few times using the relation above as a recursion.

Fix starting and final points $p_1$ and $p_2$, and consider different paths in a form: $$p_1 + (p_2 - p_1)\lambda + \vec{a}\sin(\pi\lambda) + \vec{b}\sin(2\pi\lambda) + \vec{c}\sin(3\pi\lambda)$$
Where $\lambda$ is the parameter along our path and $\vec{a},\vec{b},\vec{c}$ are 2D vectors, that parametrize it.

Take some initial parameters $(\vec{a},\vec{b},\vec{c})$ and calculate the action along the path by means of Maupertuis' principle.

Calculate the action for $(\vec{a}',\vec{b}',\vec{c}')$ parameters. If action becomes smaller -- replace the parameters with new values $\vec{a}=\vec{a}',\vec{b}=\vec{b}',\vec{c}=\vec{c}'$.

Goto step 4.

Here is what I've got in the end:

Here $\alpha = -200, p_1 = (0,-5)$ and $p_2 = (0.17,-0.17)$.
Numbers on top are: left ("Шаг") -- is a step number in the simulation, right ("Действие") -- is the value of the Maupertuis' action.

Red and green lines are real trajectories in the potential and the black line is my "test trajectory". So one can see that the simple random walk in parameter space can find some of the real paths of the body.